Multiview Subspace Clustering via Tensorial t-Product Representation

Abstract

The ubiquitous information from multiple-view data, as well as the complementary information among different views, is usually beneficial for various tasks, for example, clustering, classification, denoising, and so on. Multiview subspace clustering is based on the fact that multiview data are generated from a latent subspace. To recover the underlying subspace structure, a successful approach adopted recently has been sparse and/or low-rank subspace clustering. Despite the fact that existing subspace clustering approaches may numerically handle multiview data, by exploring all possible pairwise correlation within views, high-order statistics that can only be captured by simultaneously utilizing all views are often overlooked. As a consequence, the clustering performance of the multiview data is compromised. To address this issue, in this paper, a novel multiview clustering method is proposed by using t-product in the third-order tensor space. First, we propose a novel tensor construction method to organize multiview tensorial data, to which the tensor-tensor product can be applied. Second, based on the circular convolution operation, multiview data can be effectively represented by a t-linear combination with sparse and low-rank penalty using ``self-expressiveness.'' Our extensive experimental results on face, object, digital image, and text data demonstrate that the proposed method outperforms the state-of-the-art methods for a range of criteria.

Essential Tensor Learning for Multi-View Spectral Clustering

“…[28] extends the LRR into multi-view subspace clustering with generalized tensor nuclear norm. Then [29] adopts the t-SVD based tensor nuclear norm for better representation, and [30] proposes the tensorial t-product representation. Zhang et al [31] jointly learns the underlying latent representation of features and the multi-view low-rank representation, and then generalize it to combine with deep neural network [32].…”

A Self-Organizing Tensor Architecture for Multi-view Clustering

“…The partial symmetric CP decomposition is then applied to extract the latent feature for the clustering [19]. • SCMV-3DT is the one of the most recent third-order tensor based multi-view clustering method proposed by [12]. By using t-product based on the circular convolution, the multi-view tensor data is reconstructed by itself with sparse and low-rank penalty.…”

“…These approaches, however, fail to explore the explicit correlations between features across multiple views. Recently, several researchers proposed the use of tensor analysis method to address multi-view clustering problems [12,13,14], and it is reported to achieve competitive performance compared with conventional multi-view clustering algorithms. However, existing methods mainly focus on the third-order tensor representation, while the higher-order structural information among all views has not been fully exploited.…”

Continuity, differentiability and semismoothness of generalized tensor functions

“…Then, by the tensor T-product, the multiplication of two third-order tensors can be effectively dealt with to obtain a new third-order tensor. With these advantages, the tensor T-product has been used in many fields, such as computer vision [15,36,43], data completion [46,47,48], image processing [20,26,38], low rank minimization and robust tensor PCA [23,24,37,39].…”